A New Model Dimension Reduction Technique Based on Finite Volume Element and Proper Orthogonal Decomposition for Solving the 2D Hyperbolic Equation

This article mainly researches the model dimension reduction in the finite volume element (FVE) method based on proper orthogonal decomposition (POD) for the two-dimensional (2D) hyperbolic equation. For this objective, an FVE method with unconditional stability and second-order temporal accuracy, and the existence, stability, and error estimates of the FVE solutions are first reviewed. Thereafter, most importantly, a new FVF model dimension reduction (FVEMDR) formulation is established by applying POD technology to lower the dimension of the vectors composed of unknown coefficients for the FVE solutions. The greatest contribution of this article is the theoretical analysis of the existence, unconditional stability, and error estimations for the FVEMDR solutions. Moreover, in computation, two sets of numerical simulations are provided to confirm the validity of the theoretical results and show the effectiveness of the FVEMRD formulation.